Critical behavior in fluids and fluid mixtures

19

Fluid Phase Equilibria, 14 (1983) lS-44 Elsevier Science Publishers B.V., Amsterdam-Printed

CRITICAL

BEHAVIOR

J. M. H. Levelt Thermophysics

in The Netherlands

IN FLUIDS AND FLUID MIXTURES

Sengers,

Division,

G. Morrison, National

and R. F. Chang

Bureau

of Standards,

Washington,

D.C. 20234

ABSTRACT The presence of large fluctuations near a critical point leads to thermodynamic anomalies not present in classical, i.e., analytic equations, such as, that of van der Claals. We introduce the concepts of universality, critical exponents, scaling laws, "field" and "density" variables, "strong" and "weak" directions. Experimental evidence for the presence of nonclassical critical behavior in fluids and fluid mixtures is presented. A scaled thermodynamic potential represents the thermodynamic data of steam, ethylene and isobutane accurately. It is valid up to 1.07 T,, and within ?30% from pc. Methods for joining it to an analytic equation are discussed. The generalization to a nonclassical description of fluid mixtures is described, and applications are given. The engineer may require the nonclassical description in custody transfer, design of supercritical power cycles and supercritical extraction. The classical approach is used here to explain peculiarities of dilute mixtures recently reported by several experimenters.

INTFODUCTION The modern years, amongst point

theory

of critical

has crystallized theoreticians

prize,

about

forms

systems.

a decade

ago,

a mathematical parameter,

of the large

compressibility.

Haals

equation

fluctuations,

for fluids they

are

consequences

of these

fluctuations

is that

interparticle fluids)

that

and

interactions;

into

meter.

Non-ionized

0378-3812/83/$03.00

ranges

no longer

phenomena

character

d of the system fluids

class,

and fluid

larger

on the precise

details

have a character

characterized

mixtures

these that are

than the in most

of the

of universality;

different by only

near normal

0 1983 Elsevier Science Publishers B.V.

ignore

anomalies

(as they are

and the dimensionality

because

of the critical

of space much

in the many

point

such as the van der

for magnets,

character

introduced the Nobel

of the fluctuations

those critical

are short-ranged

depend

granted

descriptions,

The striking

over

of the critical

approach

near a critical

theory

to deal with

one universality

the dimensionality

classical

the mean-field

critical

the nature

the effects

predominant

If the forces

is, they are of identical

be grouped

Since

fluctuations.

will

the past twenty has been reached

he was recently

for describing

become

inadequate

spacings.

regarding

and for which

they extend

the phenomena

molecular

which

during

consensus

The renormalization-group

basis

of the order

developed

that a broad

and exnerimentalists

in many different

by Wilson

behavior,

to the point

systems

that

can

two parameters,

n of its order critical

points

parabelong

20 in the

universality

houses

uniaxial

disorder

phase

detailed

and

systems,

paper

show

critical from

theoretical

one-component

to

fluid

component some

on

the

a recent

CRITICAL

The cal

Helmholtz

P*=P/Pc

reduced

pressure terms

of

of

dilute

related,

of

the

departures will how

critical

have

be the

anomalies

been

extended

classical

paper

least

the

("scaled")

the

mixtures, at

characteristic

occur,

anomalies

The

engineer.

pressure

van

density

reduced

der and

critical

and

quadratic

The linear

isochore. a

Pij

and

only

order,

the

shown

approach.

in one

will

and

close

an area

in part,

index

two-

with

in which

to

the

aT*=T*-1.

AP*, namely

reduced

there

interest

Eq.

(2)

the

scaled

(1)

density

absent

on

vapor

pressure

the

(1973), equation with

(Ao*)~

because

right-hand curve

of

con-

terms

have

Eq.

P,,*(T) and

the

(1)

rearrangement

of

state:

a scaling

is a function

temperature

the

the

side

x=(~T*)/(ap*)~ of

the

of

a slight of

T*=T/T,, of

value;

density-dependent

terms

the

and

to derivatives

alone

is an example by

reduced

are

classicritical

,

...

related

the

write

a critical-point

Sengers for

+

we may

for

at

c denotes

{1+41/P30)x)

if divided

the

potential

be expanded

Thus

lowest-order

by Levelt

= P30(AP*)3

variable,

p*=p/pc, constants,

the

chemical all

+ P30(P*_l)3

density

expression

the can

temperature.

density-independent

As

and

the

in the

"scaled"

difference

are

and

Waals,

pressure,

point;

= P*-Pd*(T*)

two)

with

critical

such This

FLUIDS

the of

the

AP*

than

that

the

(1) yields

a~*=~*-1

the

(T*-lli + Pll(P*-l)(T*-1)

Eq.

pressure

to

the

working

descriptions

a

of

"scalina".

demonstrated

accuracy,

contrasted

of the

interest,

energy,

as

is the

to

actual

scaled

and

behavior

of

of criticality;

critical

Here

concern

into

of

nature

be

yielded

behavior

departures

then

also order-

behavior free

retained_

represent,

these

which

temperature;

linear

ditions been

at

of

series

‘* = ’ + I ‘oi the

way

be explained

critical

such

in Taylor

where

The

It will

with

have

show

The

these

experimental

IN ONE-COMPONENT

critical

equations,

point

transformed

been

name

indeed

systems.

in which

have

to within

do

class

alloys

methods

by the

mixtures

which

A-B

thermodynamic

goes

Ising-like

model,

and

extraction.

BEHAVIOR

Classical

also

fluid

Ising

critical

examples.

surge

in supercritical

the

ranges

be addressed are

of

Practical

will

fluids

been

the

fluids.

will

remarks

has

of

mixtures

question

renormalization-group

for

and

describe,

of

The

and

predicted

concepts that

3-dimensional antiferromagnets,

description

fluids

theory,

in a number

equations

The

which

anomalies

the and

description

that

classical

shown

of

transitions.

accurate

Ising-like will

class

ferromagnets

of

x=*T*/(~p*)~.

.

(2)

law: only

the one

From

reduced (rather

Eq.

(2)

21 On the critical isotherm where

we derive the following asymptotic power laws. AT*=O and PgP c, we have: lAP*j=IP*-l/ = P30jA~I &, with 6=3

.

(3a)

On the critical isochore, where Ap*=O, we have (aP*/ap*),, = P,, IAT*I~,

with r=l

.

(3b)

This equation implies that the compressibility

KT, which is proportional to

(aP*/aP*)T, diverges as IAT*j-', which is a strong divergence.

The expansion

coefficient CXP and the specific heat at constant pressure Cp diverge strongly, like KT, as follows from considerations

of thermodynamic

stability [see, for

instance, Rowlinson's book (1982)]. Coexisting densities are obtained by setting the pressure equal to the vapor pressure for AT*
=

The term in curly brackets in Eq. (2) is to be equal to

It follows that

(P,,~P~~)~AT*I'/~,

with

@l/2

.

(3c)

Finally, the analyticity of the Helmholtz free energy implies that the second derivative CV, the specific heat at constant volume, is finite.

Thus, on the

critical isochore,

r

D) P “field”

P

field’

Y

, “density”

T “field”

Fig. 1. Pressure versus density isotherms (----_), the critical isotherm (---), the critical isochore (------), the critical point (0) and the coexistence curve ( -) in (a) P-P space and (b) P-T space. The paths along which the critical exponents a, 8, y and 6 are defined have been indicated.

22 C *

=

V

/AT*I-~

with

Equations

(3a-d)

the

indicated

paths

Nonclassical

each

the

in

Figs.

critical

phase

producing fluctuation, curve

is

tions. 1900’s

flatter

can than

Scientists that

exponents

and

approach

and their

classical

values

along

lb.

that

critical

conditions,

because

two

phases

no longer

such

fluid

la

substantial

When the

they

critical

behavior phases

become

them.

(3d)

define

When coexisting in

.

a=0

of

can

low

transform

given

by the

curves

have

density

in

free

each

theory, have

indeed

an approximately

which

fluctuations energy

other

Consequently,

classical (1900)

the cost

into

be distinguished.

as Verschaffelt

coexistence

the

in

by spontaneous the

coexistence

ignores

known

since

cubic

shape,

fluctuathe

early

as

ETHYLENE, y

-T

hi’.

1 INDICATES

I _

1

I _

2

T-T,

01

.0.5 MlLLlBA

I

I

I

5

10

20

PC

labeled 1,2,3,4;were measured within +l% of Fig. 2. Four isochores of ethylene, (1980). Isothermal pressure differences the critical density by Hastings et al. between isochores 1 and 3, 2 and 4, and 2 and 3, are proportional to KT-~ and are The slopes of the straight lines provide plotted on a double-logarithmic scale. estimates of the critical exponent y.

23

1

_ ,p-0.521g/cm3 I

Cal/molK

Levelt

Sengers

showed

(1973,

1976).

The flattening

of the critical

lead to departures

from the classical

in the other

critical

instead

of

the value which

Most

known

volume,

the most

in argon

of 1976.

thermodynamic zero weakly

speed

to

the speed

of sound,

equations

of state,

Whether I

I

I

140

150

160

Fig. 3. The specific heat at constant volume of argon diverges weakly at the critical point, according to Voronel (1976). density

and as much

of 2% from Tc and 20% steam

from pc.

in the 50-parameter

constrained

tend to overshoot by a few degrees

steam

1980)

even

the critical

good-quality

and specific

multiparameter Webb and

exceeding

1% in

heat CR .in the range

the representation

et al.

temper-

C for Tc near room

lead to errors

instance,

not

at the critical

Rubin may

by Keenan

depends

equations

like that of Benedict,

used

as

(1979,

to pay atten-

equations

[See, for

equation

in

in classical

anomalies

Classical

point

as 25% in compressibility

namely "dip"

experimentally,

needs

nonclassical

ature

temperature;

This

and Kalyanov

the engineer

specifically

the approach

such as van der b!aals's,

on the application.

T, K

see his

point,

not present

of

in Fig. 4.

tion to these

1

heat at

has a non-

in super-critical

and displayed

predicts

the anomaly

of sound will

observed

by Erokhin

to

1900's.

indication

by Voronel,

]aT*lo".

has been repeatedly

130

theory

direct

at the critical

for instance

equal

As a consequence,

proportionally

measured

being

free energy

et

exponent

the early

Fig. 3 displays

CV measured review

in Fig. 2 of near-

of CV, the specific

the Helmholtz

analyticity.

than

since

5/4

by Hastings

nonclassical

divergence

constant that

to 5, rather

a fact

remarkably,

a weak

about

is shown

as measured

values

y exceeds

The critical-isotherm

6 is close 3, again

y equals

the compressibility

ethylene

(1980).

temperature

Thus the

the exponent

1 substantially

displays

critical al.

exponents.

exponent That

1.

studies

of this curve

and the depression

compressibility

/

in her historic

(1969).]

of Co of

24

l 653.23 ~666.23 A 673.23 x 698.93

500-

450

K K K K

J

-

c E _ u”

400-

350 -

I 25

I

30020

I 1 30 35 Pressure, MPa

I

40

45

Fig. 4. The speed of sound of supercritical steam according to measurements by Erokhin shows a minimum related to the weak divergence of C The full curves are predicted values according to a scaled equation by Leve Y‘t Sengers et al. (1983a). Densities

and

Consider variables

are

etc.,

are

P,".

are

property

strong

energy

U as

extensive

different intensive;

and

and,

they a

Wheeler

(1957)

Properties a density boundary

= (-l/v)(av/aP),

lb).

density.

a field,

These

constant

density

slope

the

are

the

of

or ap

of

with

the

being

constant

that

of

density

T,

S/V,

P,

first

type

terminology

of

of

thermodynamics of critical 4 He, developed superfluid

that

both

properties with

vapor

gives

like

respect

boundary the

are

intersects

strongly,

nonclassical

phase

U/Ni.

anomby

(1961).

hand,

a density and

The

in the

These

.

as

properties

= (l/V)(aV/aT),

diverge

other

classical

isochore path

of

such

phases.

"field",

the

that

V, Nl...

conjugate

in a direction

the

derivatives

is confluent

a

S,

properties

in coexisting

Fairbanks

derivatives On

In both

critical

and

variables

Their

fashioned after

Buckingham

descriptions.

along

who

KT

(Fig.

its

specific

second

by

KS=C-l/V)(aV/aP)S

of

the

mixtures

to

of

identical

and

constant

entiation

are

(1970)

respect

anomalies

phases.

like

with

nonclassical

weak

in general,

"density",

in multicomponent

Pippard

and

a function

in coexisting

is called

Griffiths alies

fields;

the

the

CV=(aU/aT)V

to a field

(Fig.

lb),

pressure results

curve.

and or

at

a path the

of

phase

in classical

description,

finite

derivatives

of

limiting Differ-

in classical

25 theory major

but produces distinction

the weak

exponent,

however,

performed

(constant

along which

anomalies

to be made

between

in nonclassical

critical

does not only depend

density-weak),

the critical

point

is approached.

ity shows a r-divergence

along

the critical

divergence Since,

along

however,

character

the critical this

isobar

"exponent

of the divergence

Thus,

is the of a critical

compressibil-

(Fig. la) but only a y/ii&

(both classically

and nonclassically).

does not affect

or weak),

is

but also on the path

the isothermal

isochore

renormalization"

(strong

This

The value

on the way the differentiation

constant

;

field-strong

theory.

exponents.

we will

the essential

not belabor

this point

further. Scaling

Laws

By far the easiest with

their

way to introduce

classical

counterpart,

reduced

chemical

potential

reduced

pressure

difference.

behavior

but the chemical

equation

of state with

lApi*/ = where

Dlap*16

h(x),

the nonclassical

The two quantities

potential

nonclassical

scaling

values

of all power

laws

compressibility the exponents

constants

and h(x)=l+x,

by letting

the functional

Since

before.

and the specific a and T cannot

in an

(4) classically.

B and 6 assume

their

form of h(x) appropriately

law, Eq. (4), can thus be seen as a compact

introduced

6 by the following

variable

Thus we write

,

law is obtained

and by choosing

The scaling

(see below).

is to begin

for the

than for the

dependent

variables.

B(AT*)/IA~*I"~

laws

have the same asymptotic

is the appropriate

o and T as independent

x =

scaling

this equation

AU*=~*-U~*(T*), rather

difference

B and D are two substance-dependent

The nonclassical

We write

Eq. (2).

the anomalous

heat can be obtained Indeed,

be independent.

from

behavior

"packaging"

of the

(4), it follows

they are related

that

to R and

relations:

2-a = .5(6+1); y = 8(6-l),

(5)

so that only two of the four are yielded

accurate

will

therefore

will

assume

values

not consider

theoretical

a = 0.110

* 0.0045

0.325

+ 0.0015

6=

(The exponent empty

unless

property

al will

these

estimates

y = 6=

region

1.241 ? 0.0020 4.82

the equation

shortly.)

of state

than those

A, = 0.498 t

theory

systems.

parameters;

and Zinn Justin

; .

(derived)

function

of Ising-like

as fitting

of Le Guillou

be introduced

other

Renormalization-group

exponents

exponents

a form for the scaling

is required:

the one-phase

; ;

independent.

for the critical

rather

has We

we

(1980). 0.020

; (6)

The scaling h(x)

should

law, Eq. (5), is

is specified.

The following

have no nonanalyticities

at the critical

point.

In addition,

in it is

26 desirable

that

for

thermodynamic

other

have

relations

these

et al. have

integrable

been

proposed,

an

higher-order

up*

= krsa

et

thermodynamic

function

variable"

two

obey

Eubank

in the

variables,

of a "contour

or all

et al.

one-phase

overcame into

variables r, raised

some

can

have

been

which

and

region

the

(1969),

All

of

are

not

difficulties physical

properties

are

to an appropriate

variable"

and

to

scaling

et al.

(1979).

these

be obtained hard

of the

by Vicentini-Missoni

(1969),

(1969)

expressions

requirements

that

instance, al.

Schofield

parametric

The

state

nonanalyticities

form.

so-called

analytic

for

closed-form

These

of

Goodwin

of a "distance

and

so that

functions.

(1969),

transformed.

products

integrable

equations

in closed

introducing are

be

Nonanalytic

fulfill.

Verbeke

h(x)

by

variables

written

as

fractional

power,

e as follows:

AT* = r(l-b202) AU*

= arBde(l-e2),

etc.

(7)

In

Fig.

5 we

of constant

4

I

I

‘/ !

e=o J

0 on

the

vapor,

coexistence

constant

+1

are

optimum

form

for

First

data.

represent prefer

fluids

not

the

seminal

and

Moldover

(1976)

the

only

in a small

asymptotic

critical

point.

of

laws

region Levelt

of

region

for

10-S

it to

in AT*.

are

valid the

Sengers

several

be of

Thus,

Hocken

around

estimated

found

most

demonstrated

(1961)

this

between

Furthermore,

Sengers

and

Secondly,

a property

experiment

that

equations

symmetry

possess.

in

state--we

form.

perfect liquid,

do

at

fluid

these of

only

not yet

describing all,

of the

nonanaly-

centered

a canonical

and

zero

all

(7) are

an equation

imply

vapor

of

of

Since

equals point,

Eqs.

side

A contour

shown.

neatly

point.

-1 on

liquid

curve.

variable

ticities

they

on the

critical

a equals

isochore,

r is also

distance

contours

fl; specifically,

the

this

several

critical

the

at the

Fig. 5. Parametric distance (r) and contour (.a) variables in the density-temperature plane for one-component fluids. The fulldrawn curves are contours of constant 0, some of which have been labeled by the appropriate 0 value. One contour of constant r (-----) is also indicated.

show

the

the

and

size

fluids, order

in order

to

of

27 apply

the scaling

choice those

of scaling

ideas we will variables

corrections

need

to asymptotic

and the use of the Wegner

Proper

scaling.

expansion

will

fulfill

goals.

Scaling

variables

The canonical of the

and Wegner

independent

direction

expansion

form we have variables

is singled

out,

preferred p=P*/T*

is the potential

and 7=-1/T*.

that of the phase

P=P*/T*

as a function

In the n-7 plane, Only one

transition

itself

sequently,

(Fig. 6).

one scaling

Con-

variable,

is chosen along this %" "weak" direction. The strong variable, intersect

If

u is chosen to IJ' the special direction.

it is chosen

at an angle

the T=Tc axis,

(Fig. 6),

aUIIHnetry between liquid

ii (which

equation;

(Up= 0)

Wegner

Fig. 6. Choice of independent scaling variables for one-component fluids, ut in the weak direction and ulr in the strong direction, in the plane of independent variables ;, T.

P=P i

reg

PSC

=

t-3

+Psc I_

ko

l”t12-a

+ P,,(A~)(AT)

. &rectfon-

as proposed

by

(1972) and worked

for fluids

by Ley-Koo

Green (1981),

out

and

we write

+ . . .

90(Uu/1Ut/CE)

+

go, g, are universal

In parametric

representation,

term.

The value

ak,l~~l~-~+*l

functions

gl(Up/Iutl~d)

of the scaling

g,, and g, are simple

of the ut in the first

the leading

P

,

The functions

exponent

part,

the

analytic!),

_

= *in + P,,(T)

a

latter

is not

the first

to-scaling

i

a classical

for the

relation

Including

in 7 and

is not to be

with

and a scaled

of a

regular

part

identified

and

The

P consists

Preg.

P(G,i)

vapor

is introduced.

potential part,

to

Wegner

correction

of the gap exponent

+

A,

is

is A, listed

uU/lutlBs.

in e2. higher

(8)

.

variable

polynomials term

,..

The

than

in Eq. (6).

that of In

28 the

regular

while

part

P,,

Applications

of

Expansion

the

(1980)

et al.

(1983a)

et al.

(1983)

range

the

potential,

of

to

been

used the

to represent heavy

1

the

is represented

asymmetric

as a polynomial

term

in T,

in Preq.

expansion

represent

for

PO(T

multiplying

Wegner

(8) has

et al.

The

of

is a constant

the

water;

applicability

.on around in a regi' equation

for

of

state

thermodynamic

critical

surface

and

by Levelt

the

case

of

the

of ethylene;

Sengers

steam

point

of steam; et al.

by Hastings

by Levelt

(1983b)

is indicated

Sengers

by Kamgar-Parsi for

in Fig.

isobutane.

7.

In

I

I

690 Y # z d k!

660

P

660

670

650

-24

640

-22 t

1

I

I

200

I

I

I

300 DENSITY,

400

I

650

b’,$

I

I

660

670

TEMPERATURE

(0)

units, (T*-1)

-0.005 The

Wegner

of 0.1% sound

the

0.07

in pressure

are

range

expansion

fitted

I

690

, K

(b)

Fig. 7. Range of applicability of in steam, (a) in P-p and scaling, reduced

I

I

660

is roughly

and

fits

the

level

the

p*

0.7

or better.

on the

the Wegner expansion, (b) in P-T space.

PVT

same

1.3

data

all

fluids

one

correction

investigated,

to namely

.

(9)

of these

Derivatives of a few

for

with

fluids

such

percent

to within

as CP,

or better

Cv and (Figs.

their the

accuracy

speed

4, 8,

9).

of The

29

I

500_

0 22.5651 v 23.5360 A 24.5166

MPa MPa MPa

I

I Sirota’s temperature scale lowered

400 Y

2

2 .

300 -

0”

x

Density,

kg/m3

Fig. 8. Prediction of the specific heat Cp of steam by the scaled equation with one Wegner correction-to-scaling, compared with the CP data measured by Sirota. After Levelt Sengers et al. (1983a).

0 640

.’

660

I

660

Temper0ture.K

Temperature.K

Temperature. K

9. Prediction of the specific heat Cv of steam by the scaled equation with Fig. one Wegner correction term, compared with the Cy data measured by Amirkhanov. After Levelt Sengers et al. (1983a).

30 Wegner

expansion

quality

thus

thermodynamic

gives data

a quantitatively of

pure

correct

fluids

in the

representation

critical

of

high-

region.

Crossover The

engineer

in a limited and

scaled

have

some

ones

to

solutions

of

co2.

functions these

Recent f for

was

able

energy

the

of

C

V

proved

reported

Helmholtz

of

for

steam,

The

and

from

A of free

Only

(1974)

steam,

"distance to zero

from far

representation

the

from

of the

an essential

the

the

critical

of to

Cv

between

their

and

analytic

the

region

for

400

-1 I

-

I 150

L___________________---_~

I 200

Density,

I I I I I I I

I 250

I 300

kg/m3

met

by et

with

practical

Levelt al.

(1983b)

of

the isobutane

partial

success.

analytic

are

from

points

the

surface and

(1983)

the

potential,

generated

Waxman

in

switching.

In constructing

both Fig. 10. A scaled equation has been fitted to the PVT data of isobutane in the range indicated by the heavy rectangle, including data points generated from a global surface valid everywhere except inside the dotted rectangle. In most of the range of overlap, Cp values derived from either surface agree to within 1%.

scaled

representing

scaled

U

recently

properties

____---____

all A,

values of

A simple approach

-

have

intermediate

Sengers

410

state,

properties

employed

420

f is

t!oolley

it is not

possible

-

of

where

,

poiLEY;

equation

and

heat

switch

origin

Although

three

430

of

ASc+(l-f)A

it.

first

specific

difficulty:

4401

the

to a scaled

a variety

A=f

analytical

were

elucidated

as

only

gradually

a classical

constructed

energy

valid

of joining

one.

in representing

energy

he noted

problem

Rowlinson

crossing

who

decays

a representation

be a difficult

(1983)

defined and

a satisfactory data

to

Helmholtz

a suitably

with

point.

difficulties

free

the

point

satisfied

Chapela

function"

by Woolley

He wrote

to obtain

IJ, and

They

critical

be

critical

emerging.

work

the

f is a function at

been

state.

not

the has

a "switch

problems.

unity

obviously

around

representations

propose

equation C" of

will

range

of

Gallagher

in a range

where

representations valid,

in the

fit.

were

included

In this

way,

a good

mesh

two

surfaces

was

of

the

31 In the supercritical

obtained. less than Some

1% in the range

problems

experimental The

remained,

thermodynamic

as a classical variables

surface

equation

that crosses

solution

BEHAVIOR

Phenomenology We begin are familiar

over

the phase

behaves

from

it.

surface

Ising-like the first

to this problem

would

lack of reliable

be to construct

can be achieved

behavior

a point

functions

(1983) constructed

correction

and

the

equation,

terms.

by Fox, to whose

of

a

to the van der Waals

Wegner

was devised

by having

nonanalytic

and Sengers

three

by

(Fig. 10).

near the critical

be appropriately

Ley-Koo

validity

factor.

Ising-like

This

such as CP differed

claim

boundary;

of crossover

that

from

properties

contributing

Recently,

incorporating

the reader

CRITICAL

away

variables.

correctly

approximate topic

near

to the problem

in the thermodynamic

the physical

while

however,

surface

derived

both representations

data was the principal

ideal solution

single

range,

where

A simpler

paper

on this

is referred. IN FLUID MIXTURES

and classical by showing

corresponding

a picture

to all engineers.

of phase

In

states behavior

of binary

Fig. 11, a P-x diagram shown

fluid mixtures

along

some isotherms

temperatures critical

P

second

first

lower

component.

of the Above

temperature

component,

curves

loops

the critical

Horizontal

tielines

coexisting

phases

connect

Fig. 12 we display

In

schematically

the classical

corresponding-states of the mixture (1890).

The plot along

pure fluid

treatment

by van der Waals

behavior

shows the

isotherms

in reduced

It is assumed at constant

coordinates.

composition

obeys

equation

The mixture,

separates

P-V

for a

that a mixture

the same reduced state.

to

point,

lies at the top of the

loop.

Fig. 11. P-x coexistence loops (-) at constant temperatures Tl--Tq. For temperatures between the critical temperatures of the two components, the loops have a critical point (0) at the top. The critical line (---) intersects the loops. A tieline is indicated.

that line.

and shrink

zero at the critical which

the

of the

the coexistence

are closed

intersect

is

at

than the

temperature

critical

that

of a fluid mixture

of

however,

into two phases

32 because

of material

before the

P/PC

x=x1

the

pure

of material in

’

=--,q

‘y’

\’

\J-“\ \/’

\\

\_

(1)

/4 \

-y,j

critical

point,

which

acceptable point

above

as

long

the

two-phase

region.

There

are

tie

in this

plot

because

phases

have

in general

the

for

composition.

obviously

the and

the

are

finite

of the

mixture

are

fluids.

I where

separates

"vv

"sv

“sv

%s

the

the

pseudocritical

the

point

properties not

in the

at

the

show

these

considerations from

One

of

of

is not

identified

found

stability,

course,

pure-fluid This

strong

are

corresponding

can,

mixture.

analoaous

one-component

&

but

than

Cpx

diverging

Mixtures

divergences,

diverge

heat

critical

these

counteroarts

for

the

Thus,

strongly

and

coefficient

point.

to their

that

KTx,

specific at

no

Classical

predicts

expansion

dPx

from

the

critical

position

is

with

the

critical

pure

fluid

considerations

mechanical

Det

as

point

that

compressibility

thus

properties

mixture phase

enters

lines

and

the the

mixture.

It is carefully first

mixture

is a pseudocritical

of the

Stability

the

of

isotherm

states. that

of

top

curve

rather

maintain

point

is

that

boundary

theory

Fig. 12. The mixture of composition x has the same reduced eouation of state as the pure fluid, but'separates at the curve of material instability (-----) rather than that of mechanical The pure-fluid instability (-). critical ooint (0). (1). Is a pseudocritical o>nt for-the mixture, for which (O), (2), is the real critical noint.

Note

critical

different

‘1’

of The

instability

12.

at the

coexistent

\\ \

-----

Fig.

curve

is reached.

shown

is not

\

fluid

curve

the

instability,

coexistence

explained

into

two

stability 2

0

in Rowlinson's

phases

because

book

(1982)

of mechanical

that

the

instability.

The

condition

is

>

(10)

I

indices

S, V denote

partial

derivatives

of the

energy

U with

respect

33 to entropy value

S or volume

0, from which

V. At the critical

it follows

also a P and CP) are strongly (a'P/aVZ)T=O

is needed

critical

isotherm.

written,

following

free energy AVV

Det I where

divergent.

10,

the determinant

The second

criticality

the stability

(.1982), in terms

reaches

the

so that KT (and, a fortiori,

to remain mechanically

For a binary mixture, Rowlinson

point,

(aP/aV)T=O,

for the fluid

A at constant

AVX

that

condition,

stable along

requirement

of the derivatives

the

may be

of the Helmholtz

temperature:

T=constant,

(11)

AVx Axx I it is to be remembered

everywhere,

except

From Eq. (11) and further criticality where

conditions

stability

derivatives,

because

laboratory.

Some notable

Observations

of strong

experiments,

anomaly since

anomalies

will,

plot of log I vs. log (T-Tc) yields

observed

it in the binary

point and Giglio dioxide

An interesting a chemical ionized

between

force

ionized

Thirdly, not diverge

although strongly,

aPx and Cpx diverge A fourth

case

(1973) found

fluid, develop

since

it

are usually in the

next.

in light-scattering I of scattered

point.

Charge

and

and Sengers

(1968)

near the consolute

it in the gas-liquid

mixture

of

point. at constant

An example

conservation

A

is steam

is by inducing being sliqhtly

and the chemical

on a path of a=O, so that

measures

light. A

line of slope yzl; White

the binary mixture

its components.

steam

observed

in pure C02. McIntyre

steam to remain

CPA, not Cpx!

Remarks

equilibrium

its properties by Gates et al.

is not a mixture.

the compressibility an exception

of a binary mixture,

is the case of critical

in general,

azeotropy

where

does KTx.

just as in the pure fluid.

in which

that of the density

a straight

near the plait

are those of the pure fluid--one (1982) notwithstanding,

be discussed

3 methylpentane-nitroethane

way to observe

at its critical

condition

liquid

and Vendramini

reaction

Thus

and along with

is not manipulated

to the intensity

this behavior

and propylene

however,

is directly

it is proportional

reported

(a2a/ax'lpT=0,

A=p2-~,, and x=x2.

strong divergences A

term.

that the

in fluid mixtures

of (ax/aA)pT

(1971)

is (aA/ax)pT=O,

These

is analytic

of the mixing

it can be derived

potentials,

the field variable

exceptions

Maccabee

carbon

mixture

of chemical

such as Cpa; apA, etc.

observed

The strong

free energy

(a~/aA)~~ is strongly divergent,

susceptibility

not directly

Helmholtz

considerations

of the binary

A equals the difference

the osmotic other

that the classical

at x=0 and x=1 in view of the properties

a strong

gradient

its local chemical

a strong density

anomaly

induced

potential

gradient

is directly

observed

by the field of gravity.

related

varies

linearly

to the diverging

with

in mixtures

is

A one-component height,

will

compressibility

KT;

34 this the

density key

liquid

gradient

to the mixture,

develops

with

an equally

ibility

quite

performed resulted

by

both strong

Straub's

KTb.

profiles

was

observed

Hocken-Moldover

in strong

chemical

et

to those

al.

density

(1983)

(1965)

mentioned varying

in mixtures in pure

gradients

the

C02.

Straub

linearly

Recent

and

with

and

N20

give

was

gas-

height,

diverging

model

He3-He4

(1965)

A binary

to the

of CO2

mixtures

(Fig.

and

earlier.

related

gradient,

for

x-

z.g

Schmidt

potentials

density

experiments

analogous

Chang

by

experiment

compress-

density

calculations

and

C02-C2H6

also

13).

*(lo-S)

.8

m4 0.0

E

.p

-.4

8

-.8

-1.6

F7'g. 13. Model calculatlgns of gravity effects in mixtures of 50, 72 and 84% . . being azeotropic. The CRH6 in CO2, within 5x10) from Tc, the second mixture are strong, several percent over a few mm height, just density gradients (The concentration gradients (-----) have a complex as in one-component fluid. After Chang et al. (1983). structure. The

fifth

extraction. maintains

the

concentration

behavior flui‘d.

case

where

Here,

the

solute's versus

as the curve

a strong

anomaly

becomes

presence

of the

chemical

potential

pressure,

near

the

of densi‘ty Yersus

solute

visible

in a second,

virtually

is that

constant.

critical

endpoint,

pressure

for

of

supercritical

generally The

shows

solid curve

the

a near-critical

phase

of

same one-component

35 Nonclassical It will

critical come

the classical behavior weak

theory

is taken

of fluid mixtures

with

pressure,

variables;

c is a field

to 1 as the mixture (1970).

is a dependent

variable

be y anomalies

earlier

just

model

concerns

by the same procedure,

the binary

variable

is a normalized

from pure component

predicted

as in the one-component

based on For the

gas-liquid

phase

5 as independent

activity

and runs from 0

shortly,

2.

See

the pressure

fluid model.)

(PI

the

fluids.

1 to pure component

to be described

by

if nonclassical

problem

for one-component

and third

which

anomalies

interesting

let us consider

(In the mixture

variable,

will

to find those

employed

temperature

varies

that the strong

The really

conceptualization,

transition

Griffiths

proceed

considerations,

sake of easier

to learn

into account. We will

anomalies.

geometrical

behavior

as no surprise

Fig. 14,

In

:w

“field”

5 “field” Fig. 14. Space of independent field variables for a binary gas-liquid mixtures. 5 is a function of the two chemical potentials. The vapor pressure curves of the two components are indicated at t,=O and c=l, respectively, while the critical line (----) connects the two critical points. In a plane c=constant the scaling behavior is that of the pure fluid; cf. Fig. 6 for the choice of scaling variables. we plot the space of independent curves

critical reason

variables,

at c=O and (;=l. The physical points.

The two-phase

as in Fig. lb, since

region

the field

with

critical

the two pure-fluid

line connects

has collapsed variables

to a surface

are equal

vapor

pressure

the two pure-fluid for the same

in coexisting

phases.

36 Critical-point

universality

demands

that

at constant

5 be that

of

scaling

constants.

Note

nonclassical

constant terms

field

of

the

approach tion

on

The

developed

the

u t,

one,

parallel

two

Derivatives

taken

that

is the

of ut,

or

just

such

predicted

by

is a third the

in the

case

line

and

behavior expect

to be

These all

surface

the

if there

Griffiths of

The

just in the

the

as

classical and

erature We been

by

PC,

and

C

of

the

several

bears on

nonclassical nonclassical

the

vapor-liquid

fluids.

by Wenzel

prediction

that

and

one

situations

azeotropy.

in pure

paper

generalized

are

show etc.,

as there

that

the

critical

in which

This

point

line was

in

densities the

same

unevent-

themselves.

Thus

mixtures.

just

to

along

manifold

in a three-component

Px1x2

relation

V

this

mixtures

two

direction,

those

a special

along

of finite,

to multi-component

weak

a

of constant

namely

surface PC,

it.

direction,

critical

by keeping

CVx,

and

path

in fluid

fluids,

to

constant,

taken

dimensional

critical

angle

as

the the

passes

coexistence coordinate

through

discussed

In

in one-

axes,

an extremum

in detail

by

(1970).

predicted

and

found

n-l

any

chosen,

direction,

second

(instead

Finally,

as

strong

surface

conspicuous

curve,

divergent

are

density

the

at

constant.

The one

fluid,

Tc,

KSx

kept

descrip-

fluid:

at an

Derivatives

parameters

readily

one

temperature

Wheeler

these

most

the

are

keeping

is obtained

along

such

Exceptional

critical

is critical

liquid-liquid

curve,

are

is only

if the

and

T,

in one-component the

uU,

a

in

nonclassical

nonclassical

is a strong

nondivergent.

fluids. or

instance,

Which

taken

other

of

is phrased

directions

strongly. by

surface.

direction

derivatives

there

component

$ and

weakly

the

The

r;, two

one-component

along

critical-line

considerations

cases,

for

the

or

that

of the

nonuniversal in terms

states

one-component

the

states).

available

in binaries

two

the

is phrased

surface

diverge

are

KTx,

the

and

It is obtained of the

also

of constant

coesistence

This

of

instance

corresponding not

that

direction

aPx,

derivatives as

second

mixture,

the

CPx,

mixtures.

constant, ful

with

direction

n-component

or

this one.

behavior

from

point.

coexistence

for

critical

corresponding Note

direction

the

fields,

classical

critical

special

weak

as

with

apart

universality

x.

in a plane

along

x is confluent

direction,

we

as

anomalous

fluid,

classical

variable,

intersects

in which

because,

line,

to the

that

pure

of a pseudocritical

in analogy

critical

direction

direction

density

not make use

does

is then

point

5, whereas

variable,

constant

that

the

the

mixture

anomalies effect type,

A dozen in this

fails

classical

is the

examples

volume.

to model curve

have

the

been

in binary

flattening

of

such

tlis work data

overshoots

near

the

observed?

mixtures,

flattened

shows the

both of the

of

curves

convincingly

top

experimental

the

coexistence

of the

can

coexistence

critical

temp-

degrees.

already

mentioned

observed

in fluid

that

the

mixtures

y-anomaly from

the

of

the

intensity

osmotic of

susceptibility

scattered

be

that

light;

has y-values

37 close to 514 have been reported ml'xtures, see, for instance, larger

than

behave

volume, like

observed of such

description,

The subtle

theory.

in a number import that

Of the weak anomalies unequivocally

The anomaly

is strikingly

flattening

studied.

I

I

point accord-

of the excess

volume

curve

cf. Scott's

to

has been

review,

but is not

sleepless.

,

only the last one has been seen

Fig. 15 displays measured

I

the specific

by Bloemen

to that of CV in argon

1

(a2HE/sx2)BT,

critical

according

triethylamine-water similar

enthalpy,

finite

in KTx, CZ,,~and Cpx

mixture

One is that the curvatures

they remain

keep engineers

in all mixtures

the liquPd-lfquid

while

critical

The fact that y is

zero at the mixture

of liquid-li'qufd mixtures, i‘t will

and in liquid-liquid (1978).

consequences.

approach

Ix-xc~'-',

by Scott

and of the excess

(a2yE/ax2)pT,

ing to the nonclassical classical

in gas-liquid

1 has some less-Mediate

of the excess which

both

the review

(Fig. 3).

1

heat of

et al.

(1981).

Fig. 15

I

I

(1) Triethylati-Water (2) Triethylati-\-Ethanol IXE =O.O7l (3)Tri&hylamine-Emoter-EthmolIXE =O.lOJ (Ll Triethyhmine-l-!euvyWater

I

cpx(~-!_)

:;

ill",

T PC)‘Fig. 15. The weak divergence of Cpx in triethylamine-water is analogous to that of C i‘n argon (Fig. 3). Addition of a small amount of a third component destroys 1 he divergence.- After Bloemen (1981). also which

shows how this anomaly

is quenched

is in this case the mole

For a vapor-liquid from the review

system

by Voronel

fraction

if another

density

of a third

component

the same quenching (1976).

phenomenon

The weak anomaly

is held constant added

to the mixture.

is displayed

of CV in ethane

in Fig. 16, is eliminated

38

22

26

30

34

38

42

46

50

54

58

T."C The weak diveraence Fig. 16. of a small amount of heptane. by the The not in

addition weak

been

of a small

anomalies

observed

KTx was

predicted

Nonclassical

fluids.

variables

and

relatively one-component in Table

The

the

easy

use

of

the

in the

cases

to Tc

component

and

are

the

proceeds

the

same

parametric there

that

were

it was

along

the

mole

fraction.

compressibility modeled,

not

have

the

increase

observable.

same

incompressible

binary,

lines

choice The

representation. is a one-to-one

of

binary

as

one-

liquid

correspondence

indicated

for

scaling is

between

the

schematically

Parametric

form

X

krBe

A

arE6e(l-e*)

P

principle at any

and

analytic

scaled

part

~.I

akr2-ap(e)

-p2 G

mixture

of

binary

AP

the

that

at constant the

-- appropriate

U

the

and

I.

PA

By

maintained

coefficient

so close

since

Analogy one

--

of mixtures tools

to model,

fluid

heptane,

by the addition (1976).

of mixtures

modeling

component

of

expansion

certainty

to occur

modeling

Nonclassical

amount

in the

with

of CV of ethane (1) is destroyed (2) -l%;(3) -3%. After Voronel

of

akr*-*[e*(l-8*)-p(e)] universality,

constant

pressure

background

are

the

representation

in Table

Only

the

non-universal

considered

to

be functions

P.

1 is valid

scale of

for

factors

pressure

the

a,

(Levelt

k

39 Sengers,

No systematic

1983c).

in liquid-liquid

mixtures

The modeling (1973)

of gas-liquid

for the mixture

and is illustrated fluid,

P=P*/T*

critical

form,

constants those

in terms

(1977)

(1983)

for the mixture this last sented

We refer

evidence

near the critical The tools mixture

near critical

for engineers.

locating

Dilute

shown

model

have

by Moldover

by Rainwater

by Chang

and Gallagher

explanation

and

et al.

(1983)

in Fig. 13 was taken

in nonazeotropic

lines

methods

for relatively

from

(1977)

of the model,

pre-

and for the

VLE data

and in azeotropic

are either

is desired

Secondly,

that

mixtures

models

subtle

many

will

models

line,

which

critical

techniques

and Wheeler

effects

detectable,

predicted

we believe

in those applications

particularly

for

Such

Andersen

of the nonclassical

be inevitable

near the critical

calculations

the existing

have been developed.

or not experimentally

models

of nonclassical do have certain

complicated

So far, no predictive

from the lattice Although

modeling

discussed

are rather

constants.

in nonclassical

may come

accurate

The procedures

the calculation.

the use of nonclassical accuracy

and SFG-C3HB,

by Moldover

First of all, these

have been developing.

in mixtures

factors.

in para-

COB-C2H4,

of the fit of near-critical

lines.

of adjustable

input to

critical

predictive (1979)

an

the

line.

a proliferation

line forms

effect

both

are thus available

behavior

drawbacks

of the excellence

in the pure

where

is described,

for the mixture

for a clear

with this model

before

serve as reduction

and C4H10-CBH,B,

to the paper

meeting

may

behavior

C02-CBHS

CH4-N2

the gravity

CO*-CBHG;

and Griffiths

to that

and 7=1/T*,

of the Griffiths-Leung

(1975)

for the mixtures

at the Asilomar

pictorial

in dilute

that

where

mixtures.

mixtures

A variety when

Variants

for the mixtures

paper.

can be obtained

with

are

et al.

by Leung

ut (Fig. 14). The nonuniversal u P' interpolated as functions of 5 between

of the variables

out by D'Arrigio

and Gallagher

a2=+*/T*

5, the thermodynamic

background

anomalies

the lines discussed

is, in analogy

The potential

of ~,=~,*/T*,

of the two pure components.

Moldover

first developed

of one of the two components

and analytic

been worked

was

of the thermodynamic

to date.

This was done along

in Fig. 14.

as a function

parameters

analysis

mixtures

3He-4He.

In the plane of constant metric

scaled

has been performed

of recent

a fluid mixture

for instance,

the poster

large and negative solubility volume,

is another some

partial

behavior

of these

results

have resulted

the critical

by Eckert

in supercritical

by invoking critical

experiments

is near

molar

et al.

volumes

extraction, remarkable

of classical

is a simple

point

in apparently

in the present of the solute.]

the topic effects. theory,

consequence

strange

behavior

of one of its component proceedings

The enhancement

of a number By means

we will

[see,

reporting

of papers

of the in this

of some pictures,

show that this

of the near-criticality

and

"remarkable"

of the host,.

40 In

Fig.

17a,

we

plot

the

volume

of

a binary

mixture

as

,-

a function

of composition

\

1

‘\

,’

\\A/’ X

0

1

Fig. 17. (a) The Vx diagram of a gas-liquid mixture at a temperature between the critical temperatures of the two components. At the critical point (e), the partial molar volumes are obtained by drawing the tangent to the coexistence curve () and intersecting it with the x=0 and x=1 axes. An isobar (---) is drawn. The excess volume VE for this isotherm-isobar is constructed and displayed separately, (b).It is large and negative because the mixture undergoes a phase transition along the isobar. at a temperature but

far

phases

below as

we draw the

soon

the

liquid

therefore, 17a,

tielines phase.

toward

(vapor)

since

sense.

should

appear

be strongly reported behavior,

isobar

both

We can (Fig.

negative

by the see

17b).

low-volume

side

Christensen

Schofield

It will emerge

of

on

the

are

Because of

group

et al.

of

the

the

for

the

the

the

in the the

dilute

The

mixtures

curve.

17a in

In

will, Fig.

at the

side.

Had

would

we

have

Griffiths-Wheeler

excess

separation, excess

two

Fig.

to zero,

situation

"densities"

into

In

region

(liquid)

analogous

phase

shrinks

two-phase

solvent,

predominately

coexistence

predict the

host

splits

solute

low-volume

how

the

is added.

tie-line

of the

an

range.

(1983).

of

system

component

enter

volume,

immediately

the

phases,

where

and

in part

second

point,

properties now

temperature a case,

coexisting

is drawn.

instead

such

of the

critical the

critical

In

amount

side

enthalpy

found,

the

solute.

connecting

be off

(1970)

above

the

The

large-volume the

of

as a small

a typical

plotted been

slightly

that

volume excess

enthalpies show

indeed

or enthalpy volumes

will

recently this

expected

Even more understand

striking

effects

this effect,

and v2 at the mixture will

obviously

obvious

A(x,V,T)

mixing

term

A,/v = A;Vx x + A;VT

(aT) *

= Aix + "

where

with

The partial behavior

v2

immediately

of v2 and the

of the Helmholtz

in which

expansion

It is readily

(11) are,

To

v,

volume

is not

of the divergence

point,

some care.

molar

of v,

expansion

volumes

found

to leading

free

of the

that the

order:

....

,

the subscripts

indicated, which

AVV

denote

is zero.

(11) equals

Here

finite.

Along

-RT P,T = __

that

, so

Thus,

strongly however,

v, now approaches

of a nonclassical

The

V,.

model.

are reported

approach

should

as was Strongly

by Eckert

Let us now speculate which

difference

with

line,

point

as x4,

their

between

be similar

about

AVV goes to zero

product

remaining

x and AT is asymptotically

v,

+

v,

+

_!$_

(13)

.

AVx approach

V,!

of v2 to infinity

of the partial noted earlier

molar

Along

the critical

is as

1~1-*'~,

while

to the

limit

volumes

by Wheeler

partial

et al. in this

the behavior

respect

we have

;

negative

at

the determinant

is approached,

Ax, diverges,

and v, does not the approach

is thus path-dependent,

dilution

the temperature On the critical

XAVx

v2 diverges

isotherm/isobar,

to the variable point,

that V2 + V, - s

X%X

respect

critical

line the relation

on this path,

with

to the pure-fluid

critical while

to x)

the critical

it follows

T-T,,

temperature.

As the pure-fluid

0.

(proportionally

differentiation

c refers

AT equals

critical

strongly

linear;

partial

and the superscript

to the pure-fluid

x=0

17a.

nature

in the determinant

quantities.

Aix + . . . .

AVx = xx

in Fig.

critical

molar

of the parti,al molar

from the classical

at the pure-fluid

occurring

by the partial

The limiting

The precise

figure.

is to be treated

derivatives

point

-m as x-4.

of v, can be derived

energy

A

critical

approach

from this

behavior

are displayed

see the construction

molar

(1972) on the basis

volumes

at infinite

volume.

of the partial

to that of the "partial

molar

molar

specific

compressibility."

heat, From

Eq. (12) it follows that along the critical line KTx, proportional to Avv-', -1 diverges as x , since AT is linear in x along this curve. According to classical theory,

therefore, a "partial molar -2 , a very strongly behave as x

compressibility,"

must

diverging

strong

divergence

the critical Wood

(1981).

point

has been noted of steam,

Although

in dilute

in recent

in their

salt

solutions

measurements

explanation

proportional

property.

The onset

to aKTx/ax, of this very

a fair distance

from

of Cpx by Smith-Magowan

on the basis

of a corresponding-

and

42 states this

argument

will

not

they cause

assumed them

that

trouble

diverges

CPx

as long

strongly

at the

as the

solution

critical

point

critical

is far

from

line,

its critical

point. In supercritical a situation that low

the

a combination

of

solvent

two

than

mixtures,

constants

and

is very

lack

nonclassical) classical recently

Treating

near

of

the

host

inconsistency

of

power.

we

have

phase,

because The

in the density

the

sharp is due

liquid being

critical

so

of

isothermally,

solvent

the

of

to

phase

a

point.

when

limit

that

point

of

is taken

the

of

are

the

by

pure

host

This

does,

mixture,

along

into the

1902).

of

state

Fig.

however,

critical

the nothing

will

effects

account,

12.

is modified

(to reflect

which

by

was

(1980),

(1901,

by a study

non-classical

taken

mixture

et al.

of

It

(preferably

dilute

Keesom

of

simple

line.

an approach

Hastings

equation

the

of

neglect

fluid

x-0

the

though

critical

Such

be grasped

isotherm). point

The pure

see

behavior

of adjustable

an accurate

x.

mixtures can

the

treat

small

authors,

means

critical

then

of

the

approach,

near

to give

and

as

critical

in the

the

host limit

to model

proliferation

classical

namely

of dilute

critical

the

The

present

classically.

those

pure

in the

is incorrect,

the

be used

is desired

course,

the

the

and

vicinity

while

the

solvent,

host.

soluble

of complication,

accuracy

accurately

of

dome

completely

mixture,

near

pure

solvent's

in principle,

treatment

idea

vicinity

of the

improve

when

states

by one

pure

can,

a middle

classical this

the

flatness

behave

to try

proposed

Unfortunately,

more

the

the

which,

is raised

being and

pressure

predictive

description

the

of

of the

pressure

solute

of

an additional

endpoint

point

phase,

drawbacks

fail

corresponding

following

in the

the

of

will

tempting

the

the

vapor

model

with

predictive,

critical

as

mixtures

Griffiths-Leung

dilute

in excess

at a critical

as

factors: in the

function

of dilute

The

to the

solubility,

increasing

Modeling

and

the

the

is present

critical

is close

of

near

solute

becomes

solubility,

sharply

the

mixture

enhancement

of the

extraction

where

to

still in the

will

lead

to

line.

Conclusions Nonclassical

behavior

critical-point equations. quite

using

for

and

such

are:

working the

pure

of no

as the

fluids

fluids

and

of

the

for

the

fluid

such

weak

as

the

to the

solvent. mixtures.

curve

an accurate

of

Others

are

and

region;

power

and

cycles;

Nonclassical In the

the

are

generation

supercritical are

KTx, quite

is required.

power

models

treatment

the canonical

near-critical

description

supercritical

of

by analytic

divergence

engineer.

coexistence

in the

is a consequence

described

when

in supercritical of

mixtures

be properly

concern

ignored

transfer

point

fluid

effects,

flatness be

and

cannot

irmnediate

cannot

custody

critical

and

nonclassical

and

isotherms,

Examples

near

Some

subtle

apparent, P-V

in fluids

fluctuations

extraction

available

of dilute

both

mixtures,

if

43 the pure host critical

is described

anomalies,

used to describe mixture

model

with

a classical

the mixture

will

become

sufficient

accuracy

corresponding-states

near the critical

to reflect argument

the nonclassical can no longer

be

line and the use of a nonclassical

unavoidable.

Acknowledgements We have Wielopolski, figures

profited

from the

and R. H. Wood.

from their work.

insights

of J. C. Wheeler,

A. V. Voronel

J. Kestin's

remarks

and J. Thoen

R. L. Scott, permitted

led to several

P.

us to use

improvements

in the

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